Scalable, quantum-accessible, and adaptive pseudorandom quantum state and pseudorandom function-like quantum state generators

TL;DR

An isometric procedure to prepare quantum states that can be arbitrarily random is presented, which naturally gives the first construction for scalable, quantum-accessible, and adaptive PRFS assuming quantum-secure one-way functions.

Abstract

We show new constructions for pseudorandom quantum states (PRS) and pseudorandom function-like quantum state (PRFS) generators satisfying scalability, which means the security parameter can be much larger than the number of qubits, quantum accessibility, which means the adversary can provide quantum input, and adaptivity, which means the adversary can query it adaptively. We present an isometric procedure to prepare quantum states that can be arbitrarily random (i.e., the trace distance from the Haar-random state can be arbitrarily small for the true random case, or the distinguishing advantage can be arbitrarily small for the pseudorandom case). This naturally gives the first construction for scalable, quantum-accessible, and adaptive PRFS assuming quantum-secure one-way functions. Compared to prior PRFS works, we use a stronger definition of quantum accessibility in which the adversary can be ancilla-assisted, i.e., the input state may not be pure and could be entangled with other quantum registers. Thus, our result also gives the first (fully) quantum-accessible PRFS. Our PRFS construction implies various primitives, including long-input PRFS, short-input PRFS, short-output PRFS, non-adaptive PRFS, and classically-accessible adaptive PRFS. This new construction may be helpful in simplifying the microcrypt zoo.

Figures (9)

• Figure 1: Our PRFS construction implies various other primitives.
• Figure 2: Random amplitudes procedure for 3 qubits.
• Figure 3: $B_f^{(t)}: \ket{z}\ket{0} \mapsto \ket{z}\ket{\theta_{t, z}}$
• Figure 4: Controlled rotation $R: \ket{\theta}\ket{0} \mapsto \ket{\theta}(\cos (2 \pi \theta) \ket{0} + \sin(2 \pi \theta)\ket{1})$, where $\theta \in (0, 1)$, presented by $\kappa$ qubits, is the angle of turn.
• Figure 5: The random amplitudes procedure for ARS/PRS.

Theorems & Definitions (58)

• Remark 1.1: Isometry vs quantum-accessibility
• Theorem 1.1: Informal
• Theorem 1.2: Informal
• Definition 3.1: Statistical distance
• Definition 3.2: Normal distribution
• Definition 3.5: Gamma distribution
• Definition 3.7: Beta distribution
• Definition 3.9: Chi-squared distribution
• Definition 3.13: Trace distance
• Definition 3.17: Diamond-norm distance Wil17